7. PROBES OF COSMIC ACCELERATION

As described in Section 4, the phenomenon of
accelerated expansion is now well established, and the dark energy density
has been determined to a precision of a few percent. However, getting at
the nature of the dark energy—by measuring its
equation-of-state parameter—is more challenging.
To illustrate, consider that for fixed
DE, a 1%
change in (constant) w translates to only a 3% (0.3%) change in
dark-energy (total) density at redshift z = 2 and only
a 0.2% change in distances to redshifts z = 1-2.

The primary effect of dark energy is on the expansion rate of the
Universe; in turn, this affects the redshift-distance relation and the
growth of structure. While dark energy has been important at recent
epochs, we expect that its effects at high redshift were very small,
since otherwise it would have been difficult for large-scale structure
to have formed (in most models). Since
DE
/ M
(1 + z)3w ~ 1 / (1 + z)3, the
redshifts of highest leverage for probing dark energy are expected to be
between a few tenths and two
[Huterer &
Turner 2001].
Four methods hold particular promise in
probing dark energy in this redshift range: type Ia supernovae, clusters
of galaxies, baryon acoustic oscillations, and weak gravitational
lensing. In this section, we describe and compare these four probes,
highlighting their complementarity in terms of both dark energy
constraints and the systematic errors to which they are
susceptible. Because of this complementarity, a multi-pronged approach
will be most effective. The goals of the next generation of dark energy
experiments, described in Section 8, are to
constrain w0 at the few percent level and
wa at the 10% level.

While our focus is on these four techniques, we also briefly discuss other
dark-energy probes, emphasizing the important supporting role of the CMB.

By providing bright, standardizable candles
[Leibundgut
2001],
type Ia supernovae constrain cosmic
acceleration through the Hubble diagram, cf., Eq. (11). The first
direct evidence for cosmic acceleration came from SNe Ia, and they have
provided the strongest constraints on the dark energy equation-of-state
parameter. At present, they are the most effective and mature
probe of dark energy.

SN Ia light curves are powered by the radioactive decays of
56Ni (at
early times) and 56Co (after a few weeks), produced in the
thermonuclear explosion of a carbon-oxygen white dwarf accreting mass from a
companion star as it approaches the Chandrasekhar mass
[Hillebrandt
& Niemeyer 2000].
The peak luminosity is determined by the
mass of 56Ni produced in the explosion
[Arnett 1982]:
if the white dwarf is fully burned, one expects ~ 0.6
M of
56Ni to be
produced. As a result, although the detailed mechanism of SN Ia explosions
remains uncertain (e.g.,
[Hoeflich 2004,
Plewa, Calder &
Lamb 2004]),
SNe Ia are expected to have similar peak luminosities. Since they are
about as bright as a typical galaxy when they peak, SNe Ia can be
observed to large distances, recommending their utility as standard
candles for cosmology.

In fact, as Fig. 12 shows, SNe Ia are not
intrinsically standard candles, with a
1 spread of order 0.3 mag
in peak B-band luminosity which would limit their
utility. However, work in the early 1990's
[1Phillips 1993]
established an
empirical correlation between SN Ia peak brightness and the rate at
which the luminosity declines with time after peak: intrinsically
brighter SNe Ia decline more slowly. After correcting for this
correlation, SNe Ia turn out to be excellent
"standardizable" candles, with a dispersion of about 15% in
peak brightness.

Figure 12.Top panel:B-band
light curves for low-redshift SNe Ia from the Calan-Tololo survey
[Hamuy et
al. 1996]
show an intrinsic scatter of ~ 0.3 mag in peak luminosity. Bottom
panel: After a one-parameter correction for the brightness-decline
correlation, the light
curves show an intrinsic dispersion of only ~ 0.15 mag. From
[Kim 2004].

Cosmological parameters are constrained by comparing distances to low-
and high-redshift SNe Ia. Operationally, since
H0dL is independent of the
Hubble parameter H0, Eq. (11) can be written as
m = 5log10[H0dL(z;
M,
DE,
w(z))] + ,
where M - 5log10(H0
Mpc) + 25 is the parameter effectively constrained by the
low-redshift SNe that anchor the Hubble diagram.

The major systematic concerns for supernova distance measurements are
errors in correcting for host-galaxy extinction and uncertainty in the
intrinsic colors of SNe Ia; luminosity evolution; and selection bias in
the low-redshift sample. For observations in two passbands, with
perfect knowledge of intrinsic SN colors or of the extinction law, one
could solve for the extinction and eliminate its effects on the distance
modulus. In practice, the combination of photometric errors, variations
in intrinsic SN colors, and uncertainties and likely variations in
host-galaxy dust properties lead to distance uncertainties even for
multi-band observations of SNe. Observations that extend into the
rest-frame near-infrared, where the effects of extinction are much
reduced, offer promise in controlling this systematic.

With respect to luminosity evolution, there is evidence that SN peak
luminosity correlates with host-galaxy type (e.g.,
[Jha, Riess &
Kirshner 2007]),
and that the mean host-galaxy environment, e.g., the star formation rate,
evolves strongly with look-back time. However,
brightness-decline-corrected SN Ia Hubble diagrams are consistent
between different galaxy types, and since the nearby Universe spans the
range of galactic environments sampled by the high-redshift SNe, one can
measure distances to high-redshift events by comparing with low-redshift
analogs. While SNe provide a number of correlated observables
(multi-band light curves and multi-epoch spectra) to constrain the
physical state of the system, insights from SN Ia theory will likely be
needed to determine if they are collectively sufficient to constrain the
mean peak luminosity at the percent level
[Hoeflich 2004].

Finally, there is concern that the low-redshift SNe currently used to
anchor the Hubble diagram and that serve as templates for fitting
distant SN light curves are a relatively small, heterogeneously selected
sample and that correlated large-scale peculiar velocities induce larger
distance errors than previously estimated
[Hui & Greene
2006].
This situation should improve in the near future once results are collected
from low-redshift SN surveys such as the Lick Observatory Supernova
Search (LOSS), the Center for Astrophysics Supernova project, the
Carnegie Supernova Project, the Nearby Supernova Factory, and the Sloan
Digital Sky Survey-II Supernova Survey.

Accounting for systematic errors, precision measurement of
w0 and wa with SNe will
require a few thousand SN Ia light curves out to redshifts z ~
1.5 to be measured with unprecedented precision and control of
systematics
[Frieman et
al. 2003].
For redshifts
z > 0.8, this will require going to space to minimize
photometric errors, to obtain uniform light-curve coverage, and to
observe in the near-infrared bands to capture the redshifted photons.

Galaxy clusters are the largest virialized objects in the Universe.
Within the context of the CDM paradigm, the number density of
cluster-sized dark matter halos as a function of redshift and halo mass
can be accurately predicted from N-body simulations
[Warren et
al. 2006].
Comparing these predictions to
large-area cluster surveys that extend to high redshift (z 1) can
provide precise constraints on the cosmic expansion history
[Wang &
Steinhardt 1998,
Haiman, Mohr &
Holder 2001].

The redshift distribution of clusters in a survey that selects clusters
according to some observable O with redshift-dependent selection
function f(O, z) is given by

(30)

where dn(z) / dM is the space density of dark halos
in comoving coordinates, and p(O|M,z) is the
mass-observable relation, the probability that a halo of mass M
at redshift z is observed as a cluster with observable property
O. The utility of this probe hinges on the ability to robustly
associate cluster observables such as X-ray luminosity or temperature,
cluster galaxy richness, Sunyaev-Zel'dovich effect flux decrement, or
weak lensing shear, with cluster mass (e.g.,
[Borgani 2006]).

The sensitivity of cluster counts to dark energy arises from two
factors: geometry, the term multiplying the integral in
Eq. (30) is the comoving volume
element; and growth of structure, dn(z)/dM
depends on the evolution of density perturbations, cf. Eq. 15. The
cluster mass function is also
determined by the primordial spectrum of density perturbations; its
near-exponential dependence upon mass is the root of the power of
clusters to probe dark energy.

Fig. 13 shows the sensitivity to the dark energy
equation-of-state parameter of the expected cluster counts for the South
Pole Telescope and the Dark Energy Survey. At modest redshift, z
< 0.6, the differences are dominated by the volume element; at higher
redshift, the counts are most sensitive to the growth rate of perturbations.

The primary systematic concerns are uncertainties in the mass-observable
relation p(O|M,z) and in the selection
function f(O,z). The strongest cosmological
constraints arise for those cluster observables that are most strongly
correlated with mass, i.e., for which
p(O|M,z) is narrow for fixed M, and
which have a well-determined selection function. There are several
independent techniques both for detecting clusters and for estimating
their masses using observable proxies. Future surveys will aim to
combine two or more of these techniques to cross-check cluster mass
estimates and thereby control systematic error. Measurement of the
spatial correlations of clusters and of the shape of the mass function
provide additional internal calibration of the mass-observable relation
[Majumdar &
Mohr 2004,
Lima & Hu 2004].

With multi-band CCD imaging, clusters can be efficiently detected as
enhancements in the surface density of early-type galaxies, and their
observed colors provide photometric redshift estimates that
substantially reduce the projection effects that plagued early optical
cluster catalogs
[Yee & Gladders
2002,
Koester et
al. 2007].
Weak lensing and dynamical studies show that cluster richness correlates
with cluster mass
[Johnston et
al. 2007]
and can
be used to statistically calibrate mass-observable relations. Most of
the cluster baryons reside in hot, X-ray emitting gas in approximate
dynamical equilibrium in the dark matter potential well. Since X-ray
luminosity is proportional to the square of the gas density, X-ray
clusters are high-contrast objects, for which the selection function is
generally well-determined. Empirically, X-ray luminosity and temperature
are both found to correlate more tightly than optical richness with
virial mass
[Arnaud 2005,
Stanek et
al. 2006].

The hot gas in clusters also Compton scatters CMB photons as they pass
through, leading to the Sunyaev-Zel'dovich effect (SZE;
[Sunyaev &
Zeldovich 1970]),
a measurable distortion of the blackbody CMB spectrum. It can be detected
for clusters out to high redshift (e.g.,
[Carlstrom,
Holder & Reese 2002]).
Since the SZE flux decrement is linear in the gas
density, it should be less sensitive to gas dynamics
[Nagai 2006,
Motl et al. 2005].
Finally, weak gravitational lensing can be used
both to detect and to infer the masses of clusters. Since lensing is
sensitive to all mass along the line of sight, projection effects are
the major concern for shear-selected cluster samples
[Hennawi
& Spergel 2005,
White, van Waerbeke
& Mackey 2002].

X-ray or SZE measurements also enable measurements of the baryonic gas
mass in clusters; in combination with the virial mass estimates
described above, this enables estimates of the baryon gas fraction,
fgasMB / Mtot. The ratio
inferred from X-ray/SZE measurements depends upon cosmological distance
because the inferred baryon mass, MBdL5/2 (X-ray) or
dL2 (SZE), and the inferred total
mass from X-ray measurements MtotdL. If clusters are representative samples of
matter, then fgas(z)
dL3/2 or 1 should
be independent of redshift and
B
/ M; this
will only be true for the correct cosmology
[Allen et al. 2007,
Rapetti &
Allen 2007].

The peaks and troughs seen in the angular power
spectrum of the CMB temperature anisotropy (see
Fig. 5) arise from
gravity-driven acoustic oscillations of the coupled photon-baryon fluid
in the early Universe. The scale of these oscillations is set by the
sound horizon at the epoch of recombination—the distance s
that sound waves in the fluid could have traveled by that time,

(31)

where the sound speed cs is determined by the
ratio of the baryon and photon energy densities. The precise measurement
of the angular scales of the acoustic peaks by WMAP has determined
s = 147 ± 2 Mpc. After recombination, the photons and
baryons decouple, and the effective sound speed of the baryons plummets
due to the loss of photon pressure; the sound waves remain imprinted in
the baryon distribution and, through gravitational interactions, in the
dark matter distribution as well. Since the sound horizon scale provides
a "standard ruler" calibrated by the CMB anisotropy,
measurement of the baryon acoustic oscillation (BAO) scale in the galaxy
distribution provides a geometric probe of the expansion history.

In the galaxy power spectrum, this scale appears as a series of
oscillations with amplitude of order 10%, more subtle than the acoustic
oscillations in the CMB because the impact of baryons on the far larger
dark matter component is small. Measuring the BAO scale from galaxy
clustering in the transverse and line-of-sight directions yields
measurements of r(z) / s and of sH(z),
respectively
[Hu & Haiman
2003,
Seo &
Eisenstein 2003,
Blake &
Glazebrook 2003].
Spectroscopic redshift surveys can probe both,
while photometric surveys are mainly sensitive to transverse
clustering. While determining these quantities with precision requires
enormous survey volumes and millions of galaxies, N-body simulations
suggest that the systematic uncertainties associated with BAO distance
scale measurements are smaller than those of other observational probes
of dark energy. Because such large numbers of galaxies are needed, BAO
measurements provide distance estimates that are coarse-grained in
redshift.

The main systematic uncertainties in the interpretation of BAO
measurements are the effects of non-linear gravitational evolution, of
scale-dependent differences between the clustering of galaxies and of
dark matter (bias), and, for spectroscopic surveys, redshift distortions
of the clustering, which can shift the BAO features. Numerical studies
to date suggest that the resulting shift of the scale of the BAO peak in
the galaxy power spectrum is at the percent level or less
[Seo & Eisenstein
2007,
Guzik, Bernstein
& Smith 2007,
Smith, Scoccimarro
& Sheth 2007],
comparable to the forecast measurement uncertainty for future surveys
but in principle predictable from high-resolution simulations.

The gravitational bending of light by structures in the Universe distorts or
shears the images of distant galaxies; see
Fig. 14.
This distortion allows the distribution of dark matter and its evolution
with time to be measured, thereby probing the influence of dark energy on
the growth of structure.

Figure 14. Cosmic shear field (white ticks)
superimposed on the projected mass distribution from a cosmological
N-body simulation: overdense regions are bright, underdense regions are
dark. Note how the shear field is correlated with the foreground mass
distribution. Figure courtesy of T. Hamana.

The statistical signal due to gravitational lensing by large-scale
structure is termed "cosmic shear." The cosmic shear field
at a point in the sky is estimated by locally averaging the shapes of
large numbers of distant galaxies. The primary statistical measure of
the cosmic shear is the shear angular power spectrum measured as a
function of source-galaxy redshift
zs. (Additional information is obtained by
measuring the correlations between shears at different redshifts or with
foreground lensing galaxies.) The shear angular power spectrum is
[Kaiser 1992,
Hu & Jain 2004]

(32)

where denotes the angular
multipole, the weight function
W(z, zs) is the efficiency for
lensing a population of source galaxies and is determined by the
distance distributions of the source and lens galaxies, and
P(k, z) is the power spectrum of
density perturbations.

As with clusters, the dark-energy sensitivity of the shear angular power
spectrum comes from two factors: geometry—the Hubble
parameter, the angular-diameter distance, and the weight functions; and
growth of structure—through the evolution of the power
spectrum of density perturbations. It is also possible to separate these
effects and extract a purely geometric probe of dark energy from the
redshift dependence of galaxy-shear correlations
[Jain &
Taylor 2003,
Bernstein
& Jain 2004].
The three-point correlation of cosmic shear is also sensitive to dark
energy
[Takada &
Jain 2004].

The statistical uncertainty in measuring the shear power spectrum on
large scales is
[Kaiser 1992]

(33)

where fsky is the fraction of sky area covered
by the survey,
2(i) is the
variance in a single component of the (two-component) shear, and
neff is the effective number density per
steradian of galaxies with well-measured shapes. The first term in
brackets, which dominates on large scales, comes from cosmic variance of
the mass distribution, and the second, shot-noise term results from both
the variance in galaxy ellipticities ("shape noise") and
from shape-measurement errors due to noise in the images.
Fig. 15
shows the dependence on the dark energy of the shear power spectrum and
an indication of the statistical errors expected for a survey such as
LSST, assuming a survey area of 15,000 sq. deg. and effective source
galaxy density of neff = 30 galaxies per
sq. arcmin.

Systematic errors in weak lensing measurements arise from a number of
sources
[Huterer et
al. 2006]:
incorrect shear estimates, uncertainties in galaxy
photometric redshift estimates, intrinsic correlations of galaxy shapes, and
theoretical uncertainties in the mass power spectrum on small scales. The
dominant cause of galaxy shape measurement error in current lensing
surveys is the anisotropy of the image point spread function (PSF)
caused by optical and CCD distortions, tracking errors, wind shake,
atmospheric refraction, etc. This error can be diagnosed since there are
geometric constraints on the shear patterns that can be produced by
lensing that are not respected by systematic effects.
A second kind of shear measurement error arises from miscalibration of the
relation between measured galaxy shape and inferred shear,
arising from inaccurate correction for the
circular blurring of galaxy images due to atmospheric seeing.
Photometric redshift errors impact shear power spectrum estimates primarily
through uncertainties in the scatter and bias of photometric redshift
estimates in redshift bins
[Huterer et
al. 2006,
Ma, Hu & Huterer
2006].
Any tendency of galaxies to align with their neighbors — or to
align with the local mass distribution — can be confused with
alignments caused by
gravitational lensing, thus biasing dark energy determinations
[Hirata &
Seljak 2004,
Heymans et
al. 2006].
Finally, uncertainties in the theoretical mass power
spectrum on small scales could complicate attempts to use the high-multipole
( several hundred)
shear power spectrum to constrain dark
energy. Fortunately, weak lensing surveys should be able to internally
constrain the impact of such effects
[Zentner, Rudd
& Hu 2007].

While the four methods discussed above have the most probative
power, a number of other methods have been proposed, offering the
possibility of additional consistency checks. The Alcock-Paczynski test
exploits the fact that the apparent shapes of intrinsically spherical
cosmic structures depend on cosmology
[Alcock &
Paczynski 1979].
Since spatial clustering is statistically isotropic, the anisotropy of
the two-point correlation function
along and transverse to the line of sight has been proposed for this test,
e.g., using the Lyman-alpha forest
[Hui, Stebbins &
Burles 1999].

The Integrated Sachs-Wolfe (ISW) effect provided a confirmation of
cosmic acceleration, cf.
Section 4.1.2. ISW impacts the large-angle
structure of the CMB anisotropy, but
low- multipoles are subject
to large cosmic variance, limiting their power. Nevertheless, ISW is of
interest because it may be able to show the imprint of large-scale
dark-energy perturbations
[Coble, Dodelson &
Frieman 1997,
Hu &
Scranton 2004].

Gravitational radiation from inspiraling binary neutron stars or black
holes can serve as "standard sirens" to measure absolute
distances. If their redshifts can be determined, then they could be used
to probe dark energy through the Hubble diagram
[Dalal et al. 2006].

Long-duration gamma-ray bursts have been proposed as standardizable
candles (e.g.,
[1452003Schaefer]),
but their
utility as cosmological distance indicators that could be competitive
with or complementary to SNe Ia has yet to be established
[Friedman &
Bloom 2005].
The angular size-redshift relation for double radio galaxies has also been
used to derive cosmological constraints that are consistent with dark
energy
[Guerra, Daly &
Wan 2000].
The optical depth for strong gravitational lensing
(multiple imaging) of QSOs or radio sources has been proposed
[Fukugita et
al. 1992]
and used (e.g.,
[Mitchell et
al. 2005,
Chae 2007])
to provide independent
evidence for dark energy, though these measurements depend on modeling
the density profiles of lens galaxies.

Polarization measurements from distant galaxy clusters in principle
provide a sensitive probe of the growth function and hence dark energy
[Cooray, Huterer
& Baumann 2004].
The relative ages of galaxies at different
redshifts, if they can be determined reliably, provide a measurement of
dz/dt and, from Eq. (13), measure the expansion history
directly
[Jimenez &
Loeb 2002].
Measurements of the abundance of lensed arcs in galaxy
clusters, if calibrated accurately, provide a probe of dark energy
[Meneghetti et
al. 2005].

As we have stressed, there is every reason to expect that at early times
dark energy was but a tiny fraction of the energy density. Big bang
nucleosynthesis and CMB anisotropy have been used to test this
prejudice, and current data already indicate that dark energy at early
times contributes no more than ~ 5% of the total energy density
[Bean, Hansen
& Melchiorri 2001,
Doran &
Robbers 2006].

While the CMB provides precise cosmological constraints,
by itself it has little power to probe dark energy (see
Fig. 17). The reason
is simple: the CMB provides a single snapshot of the Universe at a time
when dark energy contributed but a tiny part of the total energy density
(a part in 109 for vacuum energy). Nonetheless, the CMB plays
a critical supporting role
by determining other cosmological parameters, such as the spatial curvature
and matter density, to high precision, thereby considerably
strengthening the power of the methods discussed above, cf.
Fig. 8. It also
provides the standard ruler for BAO measurements. Data from the
Planck CMB mission, scheduled for launch in 2008, will complement those
from dark energy surveys. If the Hubble parameter can be directly
measured to better than a few percent, in combination with Planck it
would also provide powerful dark energy constraints
[Hu 2005].

In Section 5.2 we discussed the
possibility that cosmic acceleration could be
explained by a modification of General Relativity on large scales. How can
we distinguish this possibility from dark energy within GR and/or test
the consistency of GR to explain cosmic acceleration?
Since modified gravity can change both the Friedmann equation and the
evolution of density perturbations,
a strategy for testing the consistency of GR and dark energy as the
explanation for acceleration is to compare results from the geometric
(expansion history) probes, e.g., SNe or BAO, with those
from the probes sensitive to the growth of structure, e.g., clusters or
weak lensing.
Differences between the two could be evidence for the need to modify GR
[Knox, Song
& Tyson 2006].
A first application of this idea to current data shows that standard GR
passes a few modest consistency tests
[Chu & Knox
2005,
Wang et al. 2007].

Four complementary
cosmological techniques have the power to probe dark energy with
high precision and thereby advance our understanding of cosmic acceleration:
Weak Gravitational Lensing (WL); type Ia supernovae (SN); Baryon Acoustic
Oscillations (BAO); and Galaxy Clusters (CL). To date, constraints upon
the dark energy equation-of-state parameter have come from combining the
results of two or more techniques, e.g., SN+BAO+CMB (see
Fig. 8)
or BAO+CMB (see Table 1), in order
to break cosmological parameter degeneracies. In the future, each of these
methods, in combination with CMB information that constrains
other cosmological parameters, will provide powerful
individual constraints on dark energy; collectively, they should
be able to approach percent-level precision on w at its
best-constrained redshift, i.e., wp (see
Fig. 17).

Table 2
summarizes these four dark energy probes, their strengths and weaknesses
and primary systematic errors. Fig. 16 gives a
visual impression of the statistical power of each of these techniques
in constraining dark energy, showing how much each of them could be
expected to improve our present knowledge of w0 and
wa in a dedicated space mission
[Albrecht et al. 2006].